Perspectivity, exchange and summand-dual-square-free modules

Abstract

Two direct-summands A and B of a right R-module M will be called strongly-perspective (s-perspective) if for any $X\subseteq M,M=A\hspace{0.17em}\oplus \hspace{0.17em}X$ iff $M=B\hspace{0.17em}\oplus \hspace{0.17em}X.$ The module M will be called s-perspective if every two isomorphic direct-summands of M are s-perspective, and a ring R will be called s-perspective if R as a right R-module is s-perspective. In this paper, we study s-perspective rings and modules, and using ideas of Khurana and Nielsen, it follows that if M is s-perspective, then M has the finite exchange property iff M is clean, iff M has the full exchange property. In particular, if M is either summand-square-free or dual-summand-square-free with the finite exchange property, then M is clean, s-perspective and has the full exchange property.

Keywords:

• Abelian and quasi-duo rings
• cancelation
• substitution and stable range 1
• exchange and clean rings
• perspective rings and modules
• regular
• unit-regular and strongly regular rings
• square-free and dual-square-free modules

2020 Mathematics Subject Classification:

• Primary: 16D40
• 16D50
• 16D60
• Secondary: 16L30
• 16L60
• 16P20
• 16P40
• 16P60